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A117934
Perfect powers (A001597) that are close, that is, between consecutive squares.
5
27, 32, 125, 128, 2187, 2197, 6434856, 6436343, 312079600999, 312079650687, 328080401001, 328080696273, 11305786504384, 11305787424768, 62854898176000, 62854912109375, 79723529268319, 79723537443243, 4550858390629024
OFFSET
1,1
COMMENTS
It appears that all pairs of close powers involve a cube. For three pairs, the other power is a 7th power. For all remaining pairs, the other power is a 5th power. If this is true, then three powers are never close.
For the first 360 terms, 176 pairs are a cube and a 5th power. The remaining four pairs are a cube and a 7th power. - Donovan Johnson, Feb 26 2011
Loxton proves that the interval [n, n+sqrt(n)] contains at most exp(40 log log n log log log n) powers for n >= 16, and hence there are at most 2*exp(40 log log n log log log n) between consecutive squares in the interval containing n. - Charles R Greathouse IV, Jun 25 2017
LINKS
Daniel J. Bernstein, Detecting perfect powers in essentially linear time, Mathematics of Computation 67 (1998), pp. 1253-1283.
John H. Loxton, Some problems involving powers of integers, Acta Arithmetica 46:2 (1986), pp. 113-123. See Bernstein, Corollary 19.5, for a correction to the proof of Theorem 1.
StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011).
EXAMPLE
27 and 32 are close because they are between 25 and 36.
MATHEMATICA
nMax=10^14; lst={}; log2Max=Ceiling[Log[2, nMax]]; bases=Table[2, {log2Max}]; powers=bases^Range[log2Max]; powers[[1]]=Infinity; currPP=1; cnt=0; While[nextPP=Min[powers]; nextPP <= nMax, pos=Flatten[Position[powers, nextPP]]; If[MemberQ[pos, 2], cnt=0, cnt++ ]; If[cnt>1, AppendTo[lst, {currPP, nextPP}]]; Do[k=pos[[i]]; bases[[k]]++; powers[[k]]=bases[[k]]^k, {i, Length[pos]}]; currPP=nextPP]; Flatten[lst]
CROSSREFS
Cf. A097056, A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).
Sequence in context: A275188 A198147 A144862 * A173136 A030134 A290842
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 03 2006
STATUS
approved